3.8.75 \(\int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx\)
Optimal. Leaf size=220 \[ -\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}}+\frac {5 \sqrt {c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt {a+b x}}+\frac {5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac {(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 220, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used =
{96, 94, 93, 208} \begin {gather*} \frac {(c+d x)^{5/2} (7 b c-3 a d)}{4 a^2 c x (a+b x)^{3/2}}+\frac {5 (c+d x)^{3/2} (7 b c-3 a d) (b c-a d)}{12 a^3 c (a+b x)^{3/2}}+\frac {5 \sqrt {c+d x} (7 b c-3 a d) (b c-a d)}{4 a^4 \sqrt {a+b x}}-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)),x]
[Out]
(5*(7*b*c - 3*a*d)*(b*c - a*d)*Sqrt[c + d*x])/(4*a^4*Sqrt[a + b*x]) + (5*(7*b*c - 3*a*d)*(b*c - a*d)*(c + d*x)
^(3/2))/(12*a^3*c*(a + b*x)^(3/2)) + ((7*b*c - 3*a*d)*(c + d*x)^(5/2))/(4*a^2*c*x*(a + b*x)^(3/2)) - (c + d*x)
^(7/2)/(2*a*c*x^2*(a + b*x)^(3/2)) - (5*Sqrt[c]*(7*b*c - 3*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(4*a^(9/2))
Rule 93
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 94
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && !(SumSimplerQ[p, 1] && !SumSimplerQ[m, 1])
Rule 96
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin {align*} \int \frac {(c+d x)^{5/2}}{x^3 (a+b x)^{5/2}} \, dx &=-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {\left (\frac {7 b c}{2}-\frac {3 a d}{2}\right ) \int \frac {(c+d x)^{5/2}}{x^2 (a+b x)^{5/2}} \, dx}{2 a c}\\ &=\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 (7 b c-3 a d) (b c-a d)) \int \frac {(c+d x)^{3/2}}{x (a+b x)^{5/2}} \, dx}{8 a^2 c}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 (7 b c-3 a d) (b c-a d)) \int \frac {\sqrt {c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^3}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 c (7 b c-3 a d) (b c-a d)) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 a^4}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}+\frac {(5 c (7 b c-3 a d) (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{4 a^4}\\ &=\frac {5 (7 b c-3 a d) (b c-a d) \sqrt {c+d x}}{4 a^4 \sqrt {a+b x}}+\frac {5 (7 b c-3 a d) (b c-a d) (c+d x)^{3/2}}{12 a^3 c (a+b x)^{3/2}}+\frac {(7 b c-3 a d) (c+d x)^{5/2}}{4 a^2 c x (a+b x)^{3/2}}-\frac {(c+d x)^{7/2}}{2 a c x^2 (a+b x)^{3/2}}-\frac {5 \sqrt {c} (7 b c-3 a d) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 159, normalized size = 0.72 \begin {gather*} \frac {\frac {1}{2} x (7 b c-3 a d) \left (3 a^{5/2} (c+d x)^{5/2}+5 x (b c-a d) \left (\sqrt {a} \sqrt {c+d x} (4 a c+a d x+3 b c x)-3 c^{3/2} (a+b x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )\right )\right )-3 a^{7/2} (c+d x)^{7/2}}{6 a^{9/2} c x^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)),x]
[Out]
(-3*a^(7/2)*(c + d*x)^(7/2) + ((7*b*c - 3*a*d)*x*(3*a^(5/2)*(c + d*x)^(5/2) + 5*(b*c - a*d)*x*(Sqrt[a]*Sqrt[c
+ d*x]*(4*a*c + 3*b*c*x + a*d*x) - 3*c^(3/2)*(a + b*x)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c +
d*x])])))/2)/(6*a^(9/2)*c*x^2*(a + b*x)^(3/2))
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IntegrateAlgebraic [B] time = 41.56, size = 4375, normalized size = 19.89 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)),x]
[Out]
(-735*b^9*c^10*Sqrt[c + d*x] + 5845*a*b^8*c^9*d*Sqrt[c + d*x] - 19772*a^2*b^7*c^8*d^2*Sqrt[c + d*x] + 37089*a^
3*b^6*c^7*d^3*Sqrt[c + d*x] - 41990*a^4*b^5*c^6*d^4*Sqrt[c + d*x] + 29095*a^5*b^4*c^5*d^5*Sqrt[c + d*x] - 1182
0*a^6*b^3*c^4*d^6*Sqrt[c + d*x] + 2467*a^7*b^2*c^3*d^7*Sqrt[c + d*x] - 179*a^8*b*c^2*d^8*Sqrt[c + d*x] + 16170
*b^9*c^9*(c + d*x)^(3/2) - 104195*a*b^8*c^8*d*(c + d*x)^(3/2) + 281953*a^2*b^7*c^7*d^2*(c + d*x)^(3/2) - 41358
3*a^3*b^6*c^6*d^3*(c + d*x)^(3/2) + 353125*a^4*b^5*c^5*d^4*(c + d*x)^(3/2) - 174425*a^5*b^4*c^4*d^5*(c + d*x)^
(3/2) + 46395*a^6*b^3*c^3*d^6*(c + d*x)^(3/2) - 5813*a^7*b^2*c^2*d^7*(c + d*x)^(3/2) + 373*a^8*b*c*d^8*(c + d*
x)^(3/2) - 92610*b^9*c^8*(c + d*x)^(5/2) + 504455*a*b^8*c^7*d*(c + d*x)^(5/2) - 1129380*a^2*b^7*c^6*d^2*(c + d
*x)^(5/2) + 1331034*a^3*b^6*c^5*d^3*(c + d*x)^(5/2) - 876900*a^4*b^5*c^4*d^4*(c + d*x)^(5/2) + 316481*a^5*b^4*
c^3*d^5*(c + d*x)^(5/2) - 58158*a^6*b^3*c^2*d^6*(c + d*x)^(5/2) + 5278*a^7*b^2*c*d^7*(c + d*x)^(5/2) - 200*a^8
*b*d^8*(c + d*x)^(5/2) + 246960*b^9*c^7*(c + d*x)^(7/2) - 1128085*a*b^8*c^6*d*(c + d*x)^(7/2) + 2059669*a^2*b^
7*c^5*d^2*(c + d*x)^(7/2) - 1903800*a^3*b^6*c^4*d^3*(c + d*x)^(7/2) + 930124*a^4*b^5*c^3*d^4*(c + d*x)^(7/2) -
231211*a^5*b^4*c^2*d^5*(c + d*x)^(7/2) + 28695*a^6*b^3*c*d^6*(c + d*x)^(7/2) - 2352*a^7*b^2*d^7*(c + d*x)^(7/
2) - 359415*b^9*c^6*(c + d*x)^(9/2) + 1345400*a*b^8*c^5*d*(c + d*x)^(9/2) - 1934630*a^2*b^7*c^4*d^2*(c + d*x)^
(9/2) + 1325040*a^3*b^6*c^3*d^3*(c + d*x)^(9/2) - 436067*a^4*b^5*c^2*d^4*(c + d*x)^(9/2) + 66464*a^5*b^4*c*d^5
*(c + d*x)^(9/2) - 6792*a^6*b^3*d^6*(c + d*x)^(9/2) + 295470*b^9*c^5*(c + d*x)^(11/2) - 872060*a*b^8*c^4*d*(c
+ d*x)^(11/2) + 926720*a^2*b^7*c^3*d^2*(c + d*x)^(11/2) - 417540*a^3*b^6*c^2*d^3*(c + d*x)^(11/2) + 74354*a^4*
b^5*c*d^4*(c + d*x)^(11/2) - 7328*a^5*b^4*d^5*(c + d*x)^(11/2) - 129360*b^9*c^4*(c + d*x)^(13/2) + 282240*a*b^
8*c^3*d*(c + d*x)^(13/2) - 194640*a^2*b^7*c^2*d^2*(c + d*x)^(13/2) + 41760*a^3*b^6*c*d^3*(c + d*x)^(13/2) - 26
88*a^4*b^5*d^4*(c + d*x)^(13/2) + 23520*b^9*c^3*(c + d*x)^(15/2) - 33600*a*b^8*c^2*d*(c + d*x)^(15/2) + 10080*
a^2*b^7*c*d^2*(c + d*x)^(15/2) + Sqrt[b/d]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]*(-105*a*b^7*c^9*d^2 + 640*a^2*b
^6*c^8*d^3 - 1641*a^3*b^5*c^7*d^4 + 2280*a^4*b^4*c^6*d^5 - 1835*a^5*b^3*c^5*d^6 + 840*a^6*b^2*c^4*d^7 - 195*a^
7*b*c^3*d^8 + 16*a^8*c^2*d^9 - 4410*b^8*c^9*d*(c + d*x) + 28840*a*b^7*c^8*d^2*(c + d*x) - 78694*a^2*b^6*c^7*d^
3*(c + d*x) + 115408*a^3*b^5*c^6*d^4*(c + d*x) - 97174*a^4*b^4*c^5*d^5*(c + d*x) + 46056*a^5*b^3*c^4*d^6*(c +
d*x) - 10954*a^6*b^2*c^3*d^7*(c + d*x) + 960*a^7*b*c^2*d^8*(c + d*x) - 32*a^8*c*d^9*(c + d*x) + 41160*b^8*c^8*
d*(c + d*x)^2 - 222950*a*b^7*c^7*d^2*(c + d*x)^2 + 493108*a^2*b^6*c^6*d^3*(c + d*x)^2 - 568387*a^3*b^5*c^5*d^4
*(c + d*x)^2 + 360110*a^4*b^4*c^4*d^5*(c + d*x)^2 - 121214*a^5*b^3*c^3*d^6*(c + d*x)^2 + 19702*a^6*b^2*c^2*d^7
*(c + d*x)^2 - 1545*a^7*b*c*d^8*(c + d*x)^2 + 16*a^8*d^9*(c + d*x)^2 - 144060*b^8*c^7*d*(c + d*x)^3 + 649740*a
*b^7*c^6*d^2*(c + d*x)^3 - 1163414*a^2*b^6*c^5*d^3*(c + d*x)^3 + 1043930*a^3*b^5*c^4*d^4*(c + d*x)^3 - 487182*
a^4*b^4*c^3*d^5*(c + d*x)^3 + 112754*a^5*b^3*c^2*d^6*(c + d*x)^3 - 12632*a^6*b^2*c*d^7*(c + d*x)^3 + 864*a^7*b
*d^8*(c + d*x)^3 + 252840*b^8*c^6*d*(c + d*x)^4 - 931945*a*b^7*c^5*d^2*(c + d*x)^4 + 1311720*a^2*b^6*c^4*d^3*(
c + d*x)^4 - 872090*a^3*b^5*c^3*d^4*(c + d*x)^4 + 275488*a^4*b^4*c^2*d^5*(c + d*x)^4 - 40149*a^5*b^3*c*d^6*(c
+ d*x)^4 + 4136*a^6*b^2*d^7*(c + d*x)^4 - 239610*b^8*c^5*d*(c + d*x)^5 + 696500*a*b^7*c^4*d^2*(c + d*x)^5 - 72
6080*a^2*b^6*c^3*d^3*(c + d*x)^5 + 319500*a^3*b^5*c^2*d^4*(c + d*x)^5 - 55910*a^4*b^4*c*d^5*(c + d*x)^5 + 5984
*a^5*b^3*d^6*(c + d*x)^5 + 117600*b^8*c^4*d*(c + d*x)^6 - 253680*a*b^7*c^3*d^2*(c + d*x)^6 + 172800*a^2*b^6*c^
2*d^3*(c + d*x)^6 - 36720*a^3*b^5*c*d^4*(c + d*x)^6 + 2688*a^4*b^4*d^5*(c + d*x)^6 - 23520*b^8*c^3*d*(c + d*x)
^7 + 33600*a*b^7*c^2*d^2*(c + d*x)^7 - 10080*a^2*b^6*c*d^3*(c + d*x)^7))/(12*a^4*b*d^3*x^2*(-(b*c) + a*d + b*(
c + d*x))*Sqrt[a - (b*c)/d + (b*(c + d*x))/d]*(a*b^4*c^4 - 4*a^2*b^3*c^3*d + 6*a^3*b^2*c^2*d^2 - 4*a^4*b*c*d^3
+ a^5*d^4 - 221*a*b^4*c^3*(c + d*x) + (49*b^5*c^4*(c + d*x))/d + 369*a^2*b^3*c^2*d*(c + d*x) - 271*a^3*b^2*c*
d^2*(c + d*x) + 74*a^4*b*d^3*(c + d*x) + 1427*a*b^4*c^2*(c + d*x)^2 - (441*b^5*c^3*(c + d*x)^2)/d - 1531*a^2*b
^3*c*d*(c + d*x)^2 + 545*a^3*b^2*d^2*(c + d*x)^2 - 2496*a*b^4*c*(c + d*x)^3 + (1176*b^5*c^2*(c + d*x)^3)/d + 1
320*a^2*b^3*d*(c + d*x)^3 + 1296*a*b^4*(c + d*x)^4 - (1232*b^5*c*(c + d*x)^4)/d + (448*b^5*(c + d*x)^5)/d) + 1
2*a^4*b*Sqrt[b/d]*d^3*x^2*(-(b*c) + a*d + b*(c + d*x))*(-42*a*b^4*c^4*Sqrt[c + d*x] + (7*b^5*c^5*Sqrt[c + d*x]
)/d + 98*a^2*b^3*c^3*d*Sqrt[c + d*x] - 112*a^3*b^2*c^2*d^2*Sqrt[c + d*x] + 63*a^4*b*c*d^3*Sqrt[c + d*x] - 14*a
^5*d^4*Sqrt[c + d*x] + 791*a*b^4*c^3*(c + d*x)^(3/2) - (182*b^5*c^4*(c + d*x)^(3/2))/d - 1281*a^2*b^3*c^2*d*(c
+ d*x)^(3/2) + 917*a^3*b^2*c*d^2*(c + d*x)^(3/2) - 245*a^4*b*d^3*(c + d*x)^(3/2) - 2877*a*b^4*c^2*(c + d*x)^(
5/2) + (903*b^5*c^3*(c + d*x)^(5/2))/d + 3045*a^2*b^3*c*d*(c + d*x)^(5/2) - 1071*a^3*b^2*d^2*(c + d*x)^(5/2) +
3648*a*b^4*c*(c + d*x)^(7/2) - (1736*b^5*c^2*(c + d*x)^(7/2))/d - 1912*a^2*b^3*d*(c + d*x)^(7/2) - 1520*a*b^4
*(c + d*x)^(9/2) + (1456*b^5*c*(c + d*x)^(9/2))/d - (448*b^5*(c + d*x)^(11/2))/d)) - (35*b^(3/2)*c^(5/2)*Sqrt[
b/d]*Sqrt[d]*ArcTanh[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]) - (Sqrt[b]*(c + d*x))/(Sqrt[a]*Sqrt[c]*Sqrt[d]) + (Sq
rt[b/d]*Sqrt[d]*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c])])/(4*a^(9/2)) - (
15*Sqrt[c]*Sqrt[b/d]*d^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]) - (Sqrt[b]*(c + d*x))/(Sqrt[a]*Sqrt[c
]*Sqrt[d]) + (Sqrt[b/d]*Sqrt[d]*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c])])
/(4*a^(5/2)*Sqrt[b]) - ((-175*Sqrt[b]*c^(11/2)*Sqrt[b/d]*d^(3/2)*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt
[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/(4*a^(7/2)) + (3465*Sqrt[b]
*c^(9/2)*Sqrt[b/d]*d^(3/2)*(c + d*x)*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d +
(b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/(4*a^(7/2)) - (2520*Sqrt[b]*c^(7/2)*Sqrt[b/d]*d^(3/2)*(
c + d*x)^2*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a
]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/a^(7/2) + (1820*Sqrt[b]*c^(5/2)*Sqrt[b/d]*d^(3/2)*(c + d*x)^3*ArcTanh[(b*c - b*(c
+ d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/a
^(7/2))/(d*x*(c^3 - 24*c^2*(c + d*x) + 80*c*(c + d*x)^2 - 64*(c + d*x)^3)) - ((225*Sqrt[b]*c^(11/2)*Sqrt[b/d]*
d^(3/2)*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*S
qrt[b]*Sqrt[c]*Sqrt[d])])/(4*a^(7/2)) - (4715*Sqrt[b]*c^(9/2)*Sqrt[b/d]*d^(3/2)*(c + d*x)*ArcTanh[(b*c - b*(c
+ d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/(4
*a^(7/2)) + (3820*Sqrt[b]*c^(7/2)*Sqrt[b/d]*d^(3/2)*(c + d*x)^2*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt[
c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/a^(7/2) - (3620*Sqrt[b]*c^(5
/2)*Sqrt[b/d]*d^(3/2)*(c + d*x)^3*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b
*(c + d*x))/d])/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d])])/a^(7/2) + (800*Sqrt[b]*c^(3/2)*Sqrt[b/d]*d^(3/2)*(c + d*x)
^4*ArcTanh[(b*c - b*(c + d*x) + Sqrt[b/d]*d*Sqrt[c + d*x]*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])/(Sqrt[a]*Sqrt[b
]*Sqrt[c]*Sqrt[d])])/a^(7/2))/(d*x*(c^3 - 24*c^2*(c + d*x) + 80*c*(c + d*x)^2 - 64*(c + d*x)^3))
________________________________________________________________________________________
fricas [A] time = 5.63, size = 659, normalized size = 3.00 \begin {gather*} \left [\frac {15 \, {\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + {\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{a}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (6 \, a^{3} c^{2} - {\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \, {\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \, {\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (7 \, b^{4} c^{2} - 10 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (7 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + {\left (7 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - 2 \, {\left (6 \, a^{3} c^{2} - {\left (105 \, b^{3} c^{2} - 115 \, a b^{2} c d + 16 \, a^{2} b d^{2}\right )} x^{3} - 2 \, {\left (70 \, a b^{2} c^{2} - 79 \, a^{2} b c d + 12 \, a^{3} d^{2}\right )} x^{2} - 3 \, {\left (7 \, a^{2} b c^{2} - 9 \, a^{3} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/x^3/(b*x+a)^(5/2),x, algorithm="fricas")
[Out]
[1/48*(15*((7*b^4*c^2 - 10*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 + 2*(7*a*b^3*c^2 - 10*a^2*b^2*c*d + 3*a^3*b*d^2)*x^3
+ (7*a^2*b^2*c^2 - 10*a^3*b*c*d + 3*a^4*d^2)*x^2)*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*
x^2 - 4*(2*a^2*c + (a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) -
4*(6*a^3*c^2 - (105*b^3*c^2 - 115*a*b^2*c*d + 16*a^2*b*d^2)*x^3 - 2*(70*a*b^2*c^2 - 79*a^2*b*c*d + 12*a^3*d^2)
*x^2 - 3*(7*a^2*b*c^2 - 9*a^3*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2), 1/24
*(15*((7*b^4*c^2 - 10*a*b^3*c*d + 3*a^2*b^2*d^2)*x^4 + 2*(7*a*b^3*c^2 - 10*a^2*b^2*c*d + 3*a^3*b*d^2)*x^3 + (7
*a^2*b^2*c^2 - 10*a^3*b*c*d + 3*a^4*d^2)*x^2)*sqrt(-c/a)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt
(d*x + c)*sqrt(-c/a)/(b*c*d*x^2 + a*c^2 + (b*c^2 + a*c*d)*x)) - 2*(6*a^3*c^2 - (105*b^3*c^2 - 115*a*b^2*c*d +
16*a^2*b*d^2)*x^3 - 2*(70*a*b^2*c^2 - 79*a^2*b*c*d + 12*a^3*d^2)*x^2 - 3*(7*a^2*b*c^2 - 9*a^3*c*d)*x)*sqrt(b*x
+ a)*sqrt(d*x + c))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)]
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giac [B] time = 11.67, size = 1694, normalized size = 7.70
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/x^3/(b*x+a)^(5/2),x, algorithm="giac")
[Out]
-5/4*(7*sqrt(b*d)*b^2*c^3*abs(b) - 10*sqrt(b*d)*a*b*c^2*d*abs(b) + 3*sqrt(b*d)*a^2*c*d^2*abs(b))*arctan(-1/2*(
b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-
a*b*c*d)*a^4*b) + 1/2*(11*sqrt(b*d)*b^8*c^6*abs(b) - 53*sqrt(b*d)*a*b^7*c^5*d*abs(b) + 102*sqrt(b*d)*a^2*b^6*c
^4*d^2*abs(b) - 98*sqrt(b*d)*a^3*b^5*c^3*d^3*abs(b) + 47*sqrt(b*d)*a^4*b^4*c^2*d^4*abs(b) - 9*sqrt(b*d)*a^5*b^
3*c*d^5*abs(b) - 33*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^6*c^5*abs(b)
+ 56*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^5*c^4*d*abs(b) + 14*sqrt
(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c^3*d^2*abs(b) - 64*sqrt(b*d)*
(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*c^2*d^3*abs(b) + 27*sqrt(b*d)*(sqrt(
b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^2*c*d^4*abs(b) + 33*sqrt(b*d)*(sqrt(b*d)*sqr
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c^4*abs(b) - 5*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sq
rt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^3*c^3*d*abs(b) + 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c +
(b*x + a)*b*d - a*b*d))^4*a^2*b^2*c^2*d^2*abs(b) - 27*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x
+ a)*b*d - a*b*d))^4*a^3*b*c*d^3*abs(b) - 11*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d -
a*b*d))^6*b^2*c^3*abs(b) + 2*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a*b*
c^2*d*abs(b) + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*c*d^2*abs(b))
/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b
^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) -
sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)^2*a^4) + 4/3*(9*sqrt(b*d)*b^6*c^5*abs(b) - 38*sqrt(b*d)*a*b^5*c^4*d*ab
s(b) + 62*sqrt(b*d)*a^2*b^4*c^3*d^2*abs(b) - 48*sqrt(b*d)*a^3*b^3*c^2*d^3*abs(b) + 17*sqrt(b*d)*a^4*b^2*c*d^4*
abs(b) - 2*sqrt(b*d)*a^5*b*d^5*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a
*b*d))^2*b^4*c^4*abs(b) + 60*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^3
*c^3*d*abs(b) - 72*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^2*c^2*d^2
*abs(b) + 36*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b*c*d^3*abs(b) -
6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*d^4*abs(b) + 9*sqrt(b*d)*(sq
rt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^2*c^3*abs(b) - 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b
*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b*c^2*d*abs(b) + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr
t(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*c*d^2*abs(b))/((b^2*c - a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c
+ (b*x + a)*b*d - a*b*d))^2)^3*a^4)
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maple [B] time = 0.03, size = 758, normalized size = 3.45 \begin {gather*} -\frac {\sqrt {d x +c}\, \left (45 a^{2} b^{2} c \,d^{2} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-150 a \,b^{3} c^{2} d \,x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+105 b^{4} c^{3} x^{4} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+90 a^{3} b c \,d^{2} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-300 a^{2} b^{2} c^{2} d \,x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+210 a \,b^{3} c^{3} x^{3} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+45 a^{4} c \,d^{2} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-150 a^{3} b \,c^{2} d \,x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+105 a^{2} b^{2} c^{3} x^{2} \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )-32 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,d^{2} x^{3}+230 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c d \,x^{3}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{3} c^{2} x^{3}-48 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} d^{2} x^{2}+316 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b c d \,x^{2}-280 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a \,b^{2} c^{2} x^{2}+54 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c d x -42 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} b \,c^{2} x +12 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{3} c^{2}\right )}{24 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {a c}\, \left (b x +a \right )^{\frac {3}{2}} a^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x+c)^(5/2)/x^3/(b*x+a)^(5/2),x)
[Out]
-1/24*(d*x+c)^(1/2)*(45*a^2*b^2*c*d^2*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)-150*
a*b^3*c^2*d*x^4*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+105*b^4*c^3*x^4*ln((a*d*x+b*c*
x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+90*a^3*b*c*d^2*x^3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x
+a)*(d*x+c))^(1/2))/x)-300*a^2*b^2*c^2*d*x^3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+2
10*a*b^3*c^3*x^3*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)+45*ln((a*d*x+b*c*x+2*a*c+2*(a
*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^4*c*d^2-150*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^
(1/2))/x)*x^2*a^3*b*c^2*d+105*ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*x^2*a^2*b^2*c^3-
32*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*d^2*x^3+230*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c*d*x^3-210
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^3*c^2*x^3-48*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*d^2*x^2+316*(a*c)^
(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c*d*x^2-280*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c^2*x^2+54*(a*c)^(1/
2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c*d*x-42*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c^2*x+12*(a*c)^(1/2)*((b*x+a
)*(d*x+c))^(1/2)*a^3*c^2)/a^4/((b*x+a)*(d*x+c))^(1/2)/x^2/(a*c)^(1/2)/(b*x+a)^(3/2)
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)^(5/2)/x^3/(b*x+a)^(5/2),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^{5/2}}{x^3\,{\left (a+b\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)),x)
[Out]
int((c + d*x)^(5/2)/(x^3*(a + b*x)^(5/2)), x)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x+c)**(5/2)/x**3/(b*x+a)**(5/2),x)
[Out]
Timed out
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